In complex analysis, the concept of residue at infinity relates to the behavior of a meromorphic function as the variable approaches infinity. To understand this, consider a meromorphic function \( f(z) \), which is a complex function that is analytic on the entire complex plane except for isolated poles.
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For a function holomorphic outside a sufficiently large disk, the residue at infinity is defined using the coordinate on the Riemann sphere:The factor comes from . Consequently the residue at infinity is the negative of the coefficient of in the large- Laurent expansion. It is not the residue of the composite function by itself.
For , the finite residue at zero is one and the residue at infinity is minus one. More generally, a meromorphic differential on the Riemann sphere has total residue zero: the finite residues and the residue at infinity cancel. Equivalently, the integral over a large counterclockwise circle is times the residue at infinity. The coordinate change explains the sign in that formula.